An Efficient Sparse Representation Algorithm
for Direction-of-Arrival Estimation
Lei SUN, Huali WANG, Guangjie XU
College of Communications Engineering, PLA Univ. of Sci.&Tech., Nanjing, China [email protected], [email protected], [email protected] Abstract. This paper presents an efficient sparse
repre-sentation approach to direction-of-arrival (DOA) esti-mation using uniform linear arrays. The proposed approach constructs the jointly sparse model in real domain by exploiting the properties of centro-Hermitian matrices. Subsequently, DOA estimation is realized via the sparse Bayesian learning (SBL) algorithm. Further, the pruning threshold of SBL is adaptively selected to speed up the basis pruning rate. Simulation results demonstrate that the proposed approach achieves an improved performance and enjoys computational efficiency as compared to the state-of-the-art l1-norm-based DOA estimators especially
in scenarios with small sample size and low signal-to-noise ratio.
Keywords
Direction-of-arrival (DOA) estimation, uniform linear array, sparse representation, sparse Bayesian learning.
1. Introduction
Direction-of-Arrival (DOA) estimation has been widely used in radar, sonar, wireless communications, and other application fields. A vast number of algorithms have been devised for the DOA estimation problem. Subspace algorithms, such as MUSIC [1], ESPRIT [2], and their variants, are well known to have high resolution capabili-ties. However, all the subspace algorithms require the exact knowledge of the array signal model and consistent esti-mate of the noise or signal subspaces. The maximum likeli-hood (ML) approach is asymptotically optimal under certain regularity conditions [3]. Unfortunately, since the ML objective function is very flat far from the true DOAs, good initial guesses for the DOAs and an exact estimate of the number of the incident sources are required to ensure the global convergence.
Recently, exploiting the inherently spatial sparsity of the incident sources, DOA estimation has been formulated in the sparse representation framework [4]-[11]. The gen-eral concept of the sparsity-based DOA estimators is to directly reconstruct the spatial spectrum of the incident sources from multiple measurement vectors (MMV)
sub-ject to sparsity constraint. In [4], the global matched filter (GMF) approach is proposed to realize the DOA estimation based on sparsely representing the beamformer samples. The so-called l1-SVD algorithm in [5] reduces the sample
size by means of the singular value decomposition (SVD), and utilizes the l1-norm penalty to enforce sparsity. The l1
-SVD algorithm has been further investigated in [6]-[8], where weighted l1-norm penalties are adopted instead.
Hyder and Mahata approximated the l0-norm by a family of
Gaussian functions, and proposed the joint l2,0
approxima-tion (JLZA) algorithm for direcapproxima-tion finding [9]. However, the global convergence of JLZA is not guaranteed since there are lots of parameters have to be properly set. The literature [10] presents a method via sparse representation of array covariance vectors provided that large sample support is available. In the scenario where uncorrelated sources impinge on a uniform linear array (ULA), Xu et al. simplified the MMV model to single measurement vector (SMV) model [11]. Generally speaking, the DOA estima-tors based on sparse representation outperform conven-tional approaches particularly in small sample size, low signal-to-noise ratio (SNR) scenarios, and rely less on the priori information of the incident source number. Nonethe-less, there are two major drawbacks of the popular l1
-norm-based approaches. First, the computational load of solving the l1-norm optimization as a second-order cone program
(SOCP) is quite expensive for some applications. Second, it is still nebulous to select the optimal regularization pa-rameter.
In this paper, we present an efficient and accurate sparse representation approach to the DOA estimation using a ULA. We construct a real-valued jointly sparse model by taking advantage of the centro-Hermitian prop-erty. The sparse Bayesian learning (SBL) algorithm [12]-[14] is adopted to realize the DOA estimation. In addition, we speed up the basis pruning rate of SBL by adaptively setting the pruning threshold according to a rough estimate of the nonzero hyper-parameters. The proposed approach achieves high estimation accuracy with a low computa-tional complexity as compared to several existing l1
-norm-based DOA estimators. Such advantages are demonstrated by simulation results.
The rest of the paper is organized as follows. Section 2 briefly reviews the formulation of the sparsity-based DOA estimators. Section 3 presents the proposed method.
Several numerical simulations are carried out in Section 4 to evaluate the performance of the proposed method. Fi-nally, Section 5 concludes the whole paper.
Notation: Throughout the present work, we use am. and a.n to denote the mth row and nth column of matrix A,
respectively. The operators E(•), (•)*, (•)T, (•)H, (•)-1, Re(•),
Im(•), diag(•), ||•||2, ||•||F, and IM denote expectation, conjugate, transpose, conjugate transpose, inverse, real part, imaginary part, diagonalization, Euclidean norm, Frobenius norm, and M×M identity matrix, respectively. j is reserved for the imaginary unit 1.
2. Problem Formulation
Consider a ULA of M sensors receives signals from K far-field uncorrelated narrowband sources. The distance between adjacent sensors is d = λ/2, where λ is the wave-length of the source. The array observation vector at time t is modeled as 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1, 2, , K k k k t s t t t t t T
x a n A θ s n (1) where sk(t) is the baseband waveform of the kth source, s(t) = [s1(t),…,sK(t)]T is the source waveform vector, n(t) is the noise vector, T denotes the number of snapshots, the vector θ = [θ1,…,θK] contains the actual DOAs of the incident sources, and the matrix A(θ) = [a(θ1),…,a(θK)] is commonly referred to as the array manifold matrix, whosekth column ( ) 1, j sin( k),..., j (M 1) sin( k) T k e e a (2)
is the steering vector of the kth source. In this paper, the source waveforms and the additive noise are assumed to be circularly symmetric, independent identically distributed (i.i.d.) complex Gaussian random processes with zero mean, and the noise at each sensor is uncorrelated with each other as well as the incident sources. Under this assumption, we obtain the following array covariance matrix:
2
{ ( ) H( )} ( ) H( )
x E t t s n M
R x x A θ R A θ I (3)
where Rs = E{s(t)sH(t)} is the source covariance matrix,
and 2
n
represents the noise variance.
To formulate the DOA estimation in the sparse repre-sentation framework, the potential spatial scope of the incident sources is first sampled to form a direction grid
1 2
[ , , , L]
Θ with L >> K denoting the length of the
grid. Correspondingly, the array manifold matrix A(θ) is generalized to an M×L over-complete dictionary B in terms
of the direction grid, as
1 2
[ ( ), ( ), , ( )]. L
B a a a (4)
Note that B is known and does not depend on θ. Suppose the grid is dense enough so that all elements of θ lie in (or,
close to) the grid, then (1) can be rewritten in an over-complete form as
X BH N (5)
where X=[x(1),…,x(T)], H=[h(1),…,h(T)], h(t) is a sparse
vector whose lth element is nonzero and equal to sk(t) if source k comes from and zero otherwise, and l N=[n(1),…,n(T)]. If the incident sources are fixed over the
observation period, it is clear that for all t the nonzero
elements of h(t) occur at the same locations, such that only K rows of H are nonzero, each corresponding to a different
source. The estimation of DOA for all sources then reduces to recoverying and determining the nonzero rows of the sparse matrix H from the snapshots matrix X, and can be expressed as the following constrained convex optimiza-tion problem:
s.t.
2,1
min H X BH F (6)
where β is a fitting error threshold that must be set by user,
and the objective ||H||2,1 is defined as
. 2,1 2 1 L l l
H h (7)which is equal to the l1-norm of the vector [||h1.||2,…,|hL.||2]T.
The indices of nonzero rows of H give the DOA estimates. It is worthwhile to note that the recovery performance of H is heavily dependent on the choice of β.
3. The Proposed Method
In this section, an efficient DOA estimator based on sparse representation is proposed. First, we formulate a real-valued jointly sparse model for the direction finding problem. Next, we develop a solution mechanism to realize the DOA estimation.
3.1 Real-Valued Jointly Sparse Model
Consider the augmented sample matrix Y=[X ПMX*ПT], where ПM is an M×M exchange matrix with ones on its anti-diagonal and zeros elsewhere. It is shown that Y is centro-Hermitian, and can be transformed to a real-valued matrix according to [15]
r HM[ M T] 2T
Y Q X Π X Π Q (8) where Q is a unitary matrix, defined as
2 2 1 1 2 n n T T n n n j j I 0 I Q 0 0 Π 0 Π (9)
and the matrix Q2n can be easily obtained from Q2n+1 by
dropping its center row and center column. Without loss of generality, we assume the amount of the sensors M = 2M′
to be even. Note that (8) incorporates forward-backward averaging, and therefore the sample size is virtually
dou-bled from T to 2T. Observe that the over-complete diction-ary B satisfies * ( ) M BΛ Π BΛ (10)
where Λ is a L×L unitary diagonal matrix whose lth
diagonal element is given by ,
( 1) sin l/ 2 1 .
j M
ll e l L
(11)
After some straightforward algebraic manipulations, we can rewrite (8) as below:
r r r r r r Re Im 2 ([ ] [ ]) 2 [ ( ) ( )] H M M T M T T Y Q BH Π B H Π N Π N Π Q B Λ H Λ H N B H N (12) where r H [ ( ), , ( )]r 1 r M L B Q BΛ a a with r
( ) 2[cos( ( 1) sin / 2), , cos( sin / 2),
sin( ( 1) sin / 2), , sin( sin / 2)] l l l T l l M M a (13) and the noise term
r 1 ' 2 1 ' 2 1 ' 2 1 ' 2 Re{ } Im{ } Im{ } Re{ } M M M M N Π N N Π N N N Π N N Π N (14) with N=[N1T, N2T]T. Hr 2[ (ReΛ H ) Im (Λ H )]
shows the same sparsity pattern as H, therefore we can achieve the DOA estimates by locating the nonzero rows of Hr. The columns of Hr and Nr still satisfy the i.i.d condition
under the circularly complex Gaussian assumption.
Before turning to solve (12), we first analyze Br from
the sparse representation viewpoint. It has been shown that the accuracy of the sparse-recovery algorithms is deter-mined by the coherence measure [16], [17] of the diction-ary matrix. For the dictiondiction-ary Br, the coherence measure is
given by r r r r 2 r 2 ( ) ( ) ( ) arg max ( ) ( ) H i l i l i l a a B a a (15)
where ar( )i and ar( )l denote the ith and lth columns of
Br, respectively. Since all the columns of Br have a
con-stant Euclidean norm M , we only have to take the nu-merator term of (15) into account. Then we get that
r r (( 1) / 2) (sin sin ) ( ) ( ) ( ) ( ) ( ) ( ) i l H i l j M H H i M M l H i l e a a a Q Q a a a (16)
where ( )ai and ( )al are the ith and lth columns of B, respectively. Considering that the Euclidean norm of each column of B is also equal to M , it is straightforward to
infer that the coherence measure of Br is the same as that of
B. As a result, the recovery accuracy of Hr is guaranteed.
3.2 DOA Estimation via SBL
We utilize the iterative SBL algorithm to solve the real-valued sparse representation problem (12). The SBL only requires ο(M2L) flops per iteration to solve (12),
pro-vided that Yr can be replaced by a M×rank(Yr) matrix r
Y such thatY Yr rT Y Yr rT . However, with the additive noise, the rank of Yr is always equal to M, thus the noise
component is also reserved inYr. Herein, to further miti-gate the infection of the additive noise, we exploit the trun-cated SVD to replace Yr:
rsv r
Y Y VD (17)
where Yr=ULV, D=[IK;0], and 0 is a (2T−K)×K zero matrix. Define Hrsv=HrVD and Nrsv=NrVD, then (12) is
simplified to the following form:
rsv r rsv rsv.
Y B H N (18)
Obviously, Hrsvand Hr have the same row support. Like the
one did in [8], we exploit the property that
rsv rsv r rsv
n y B h (19)
is an additive real Gaussian random vector with zero mean and covariance matrix rn 2
n M
R I , where nrsv, yrsv, hrsv
corresponds to the same column in Nrsv, Yrsv, Hrsv,
respec-tively. Then yrsv can be modeled through a Gaussian
likeli-hood function: rsv r rsv rsv rsv 2 2 2 2 2 / 2 1 ( | , ) exp( ). 2 (2 ) n M n n p y B h y h (20)
In a probabilistic perspective, the sparse representa-tion problem can be treated as finding hrsv and to maxi-2n mize the posterior probability p(hrsv,2n|yrsv) under
spar-sity constraint. Regarding the SBL algorithm, sparspar-sity is enforced via invoking a Gaussian prior over each element of hrsv. By combining each of these priors, we have
rsv rsv 2 1 ( ) 1 ( | ) exp( ) 2 2 L l l l l h p
h γ (21)where γ=[γ1,…,γL]T is a hyper-parameter vector with γl controlling the variance of the corresponding element rsv
l
h . If γl = 0, it means the hlrsv will be zero. If γl > 0, it indicates a nonzero rsv
l
h whose magnitude depends on γl. Using the Bayesian inference, the posterior distribution over hrsv:
rsv rsv rsv rsv rsv rsv | , , 2 2 2 ( , ) ( | ) ( | , ) ( | ) n n n p p p p y h h γ h y γ y γ (22)
turns out to be an analytically multivariate Gaussian distri-bution with the mean and covariance given by
r yr rsv r yr r 1 1 T T u ΓB Σ y Σ Γ ΓB Σ B Γ (23)
where Γdiag( )γ andΣyr n M2I B ΓBr Tr . The
poste-rior mean u is enlisted as a point estimate of hrsv. As seen
in (23), the recovery of hrsv reduces to estimate the
un-known γ and 2
n
. By summing up the contributions of the individual column rsv
.k
y , we finally get the following mar-ginal log-likelihood function:
. log , log , rsv . 1 yr 1 rsv . yr 1 2 rsv . 2 k K k T k K k n k n K p L y Σ y Σ γ y γ
(24)An iterative type-II ML procedure [12], using the EM algorithm, is employed to minimize (24) with respect to γ
and 2
n
. The E-step is achieved by calculating the posterior moment using (23). For the M-step, the hyper-parameter vector γ and the noise variance are updated: 2n
new rsv r new , 2 . 2 1 2 2 1 1 1,..., (1 ) ( ) ( ) l l l ll F n L l ll l l L K K M L
u Y B U (25)where the matrix U = [u.1,u.2,…,u.K] is comprised with the
posterior mean vector corresponding to each column of Hrsv, and Σll denotes the lth diagonal element of the poste-rior covariance matrix Σ. When a convergence criterion is satisfied, we attain the estimates of the sparse γ and noise variance 2
n
. The DOA estimation problem can be immedi-ately accomplished by deciding the locations of the rela-tively K large values of γ.
3.3 Discussions
The noise variance balances the recovery sparsity and accuracy of the SBL algorithm. It plays an important role like β in (6). However, it is demonstrated that the explicit learning rule of noise variance in (25) is not robust in strongly noisy cases. Based on the connection between the singular value and eigenvalue, we can get a maximum likelihood estimate (denoted by ˆ2
n ) of 2 n :
M K m m n K M T 1 2 2 2 1 ˆ (26)where λm is the mth singular value of Yr. As the authors
recommended in [14], we incorporate fixed ˆ2
n
into the update rules of SBL.
Since a sparse prior is invoked over each row of Hrsv,
it is observed that many elements of γ tend to zero in the
iteration process, and only a relatively small set of γ, which correspond to the incident sources, remain relatively large. A small fixed threshold (e.g., 10−5) is suggested to be set
such that, when any element of γ is below the threshold, it is pruned from the model. Herein, we propose to adaptively set the threshold according to the average value of the nonzero elements of γ. When the SBL converges, it is shown that γl satisfies
rsv 2 . . 2 1 l K l H (27) Note that rsv 2 . 2 l
H reflects 2TK times the average power of the source from [8]. Straightforwardly, we have l
total 1 2 2 1 2 2 rsv 1 2 ˆ 2 1 1 P H T T M K M m n m L l l L l l
(28)where Ptotal denotes the average total power of all incident
sources. Since only K elements of γ associated with the incident sources are nonzero, then:
total
P
min 2T / K max
(29)
where γmin and γmax denote the minimal and maximal
non-zero elements of γ, respectively. It is rational to assume that the difference between γmin and γmax is modest, thus γa=2TPtotal/K can be treated as the average value of nonzero
elements of γ. Therefore, we set the small-magnitude elements of γ to zero using the thresholding operation:
a if 10log( , otherwise 0, / ) ( )i i i H (30)
after one iteration. As a consequence, elements of γ that are more than τ dB down from γa, as well as the corresponding
columns in Br, are pruned from the model. We shall point
out that this simple modification leads to an improved basis pruning rate in practice.
In the proposed method, we use the information about the number of sources K. Practically, this information is often estimated via the classical AIC or MDL criteria [18]. It is known that incorrect estimation of the number of the incident sources degrades the performance of the l1-SVD
algorithm. For the proposed method, this degradation is alleviated, since the forward-backward averaging improves the estimation accuracy of K.
Remark: Considering the computational complexity
of the proposed method, it only requires real additions to transform the complex sample matrix into a real one [15], and the calculation of SVD is about ο(M3). Note that L >> M > K, then the recovery procedure based on SBL
dominates the complexity analysis. Since the computa-tional load of SBL per iteration is fixed, the overall com-plexity rests with the number of iterations. In moderate SNR cases, the SBL takes less than 25 iterations provided
that both the noise variance and the pruning threshold are set as we suggested. Regarding the l1-SVD algorithm
calculated by SOCP software package SeDuMi [19], the computational cost per iteration is ο((KL)3), which denotes
complex operations, and the theoretical worst-case bound on the number of iterations is ο((KL)0.5). For the l
1-ACCV
algorithm in [11], the factor K is removed, leading to a considerable reduction compared respect to l1-SVD.
Based on the previous analysis, we can conclude that the proposed method is more efficient than both l1-SVD and l1-ACCV.
4. Simulations
In this section, we carry out numerical simulations to evaluate the proposed method. We also compare it with the
l1-SVD in [5], the NSW-l1 in [6], and the l1-ACCV in [11].
In all the simulations, a 10-element ULA with half-wave-length spacing is used. The root-mean-square-error (RMSE) of the DOA estimates obtained with 500 Monte Carlo runs is defined as
K k n k kn K 1 500 1 2 ˆ 500 1 RMSE (31)where ˆkn represents the estimate of θk in the nth trial. Each narrowband source is generated from a zero mean Gaus-sian distribution. The additive GausGaus-sian noise is assumed to be white, both spatially and temporally, and uncorrelated with the sources. The SNR is defined as 10 log( 2/ 2)
s n with 2 s and 2 n
denoting the source and noise power, respectively. The number of incident sources is estimated by MDL criterion unless otherwise stated. The spatial grid is uniform in the range −89° to 90° with 1° interval, which means that L = 180. We set the pruning threshold of the proposed method to −20 dB. The confidence interval is set to 0.99 for both l1-SVD and NSW-l1. The regularization
parameter of l1-ACCV is set according to the suggested
value in [11].
Suppose four equal-power uncorrelated sources impinge on the array from [−20.4°, −12°, 15.6°, 30°]. The SNR is −5 dB, and 100 snapshots are collected. Fig. 1 depicts the normalized spatial spectra of various algorithms. Although the DOAs of the first and the fourth signal are not in the spatial grid, it is observed from Fig. 1 that all the algorithms can approximately locate them. However, spurious peaks exist in the spectrum of l1-SVD.
To see more clearly the capability of the proposed method in resolving closely spaced sources, we plot the bias of the DOA estimates against angle separation be-tween two sources in Fig. 2. We simulate two equal-power uncorrelated sources impinging from −20° and −20°+Δθ, where Δθ varies from 2° to 30°. The SNR is 0 dB and the number of snapshots is 100. It can be seen that the pro-posed method shows the best performance. The reason is
that the SBL provides a tighter approximation to the
l0-norm than the l1-norm [20], [21].
-40 -30 -20 -10 0 10 20 30 40 -160 -140 -120 -100 -80 -60 -40 -20 0 DOA(°) Po w e r( d B ) Proposed method L1-ACCV L1-SVD NSW-L1
Fig. 1. Spatial spectra obtained by different algorithms.
5 10 15 20 25 30 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Angle Seperation(°) Bia s( °) Proposed method L1-ACCV L1-SVD NSW-L1
Fig. 2. Bias of the DOA estimates versus angle separation.
-10 -8 -6 -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 SNR(dB) RM SE( °) Proposed method L1-ACCV L1-SVD NSW-L1
Fig. 3. RMSE of the DOA estimates against input SNR with
200 snapshots.
Next, the performance of these four algorithms is sta-tistically compared for incident sources of various SNRs and sample sizes. Here we simulate four equal-power
un-correlated sources located at [−20.4°, −12°, 15.6°, 30°]. First, we keep the snapshots fixed at 200 and vary the SNR of all sources from −10 dB to 8 dB. Then the SNR is fixed at −5 dB and the number of snapshots is varied from 50 to 300. The RMSE performance is depicted in Fig. 3 and Fig. 4, respectively. It can be seen from the two figures that the proposed method outperforms the other three algo-rithms in the low SNR and small sample size cases.
50 100 150 200 250 300 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Number of Snapshots RM SE( °) Proposed method L1-ACCV L1-SVD NSW-L1
Fig. 4. RMSE of the DOA estimates against the number of
snapshots with −5 dB SNR.
Finally, we present an evaluation of the computational complexity using the TIC and TOC instruction in MAT-LAB. Since the NSW-l1 performs the same computational
complexity as the l1-SVD, we only choose l1-SVD for the
comparison. Four equal-power uncorrelated sources are simulated to impinge on the array from [−20°, −12°, 10°, 30°].The number of incident sources is assumed to be known beforehand. The SNR is set at −5 dB and 0 dB, and the sample size is set at 50, 100 and 200. For each SNR-sample size pair, the computation time of these three algo-rithms are averaged over 500 trials, and the results are illustrated in Tab. 1.
Computation Time (sec) SNR
(dB)
Sample
Size l1-SVD l1-ACCV Proposed method
50 0.436 0.275 0.050 100 0.448 0.276 0.032 −5 200 0.464 0.278 0.021 50 0.463 0.270 0.026 100 0.484 0.272 0.017 0 200 0.506 0.279 0.014
Tab. 1. Computation time comparison of various algorithms.
As can be seen in Tab. 1, the computation time of the proposed method is much smaller than those two l1-norm
based algorithms. Furthermore, the computation time of
l1-SVD and l1-ACCV increases slightly with sample size. It
is worth noting the computation time of the proposed method turns out to be reduced when the SNR and sample size increase. Consequently, the proposed method should be preferred for practical applications.
5. Conclusion
In this paper, we propose a DOA estimator based on SBL with real-valued processing, which exploits the centro-Hermitian property of the ULA. An approach to adaptively choose the pruning threshold is also presented to speed up the basis pruning rate of the SBL. The simula-tion results show that the proposed method provides an improved performance in comparison with the current state-of-the-art l1-norm-based DOA estimators with
re-duced computational complexity. It is worthwhile to note that the proposed approach is not confined to ULA, but can be applied to arrays with centro-symmetric configuration.
Acknowledgements
This work is supported by the National Natural Sci-ence Foundation under Grant 61271354 of the People’s Republic of China. The authors are grateful to D. Maliou-tov for providing the source codes of l1-SVD.
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About Authors ...
Lei SUN was born in Anhui, China. He received his
Bachelor’s degree in System Engineering from the PLA University of Science and Technology, Nanjing, in 2007. His research interests include array signal processing and sparse representation. Currently, he is working towards the Ph.D degree in the PLA University of Science and Tech-nology, Nanjing, China.
HuaLi WANG was born in Zhejiang, China. He received
his Ph.D. degree in fusee technology from the University of Science and Technology, Nanjing, China, in 1997. His research interests include array signal processing and com-pressive sensing. Currently, he is a professor with the PLA University of Science and Technology, Nanjing, China. Prof. Wang is a member of the IEEE.
Guangjie XU was born in Jiangsu, China. He received his Bachelor’s degree in Electronic Engineering from the PLA University of Science and Technology, Nanjing, in 2009. His research interests include compressive sampling and signal detection. Currently, he is working towards the Ph.D degree in the PLA University of Science and Technology, Nanjing, China.
